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Research Papers: Conduction

# Simple Explicit Equations for Transient Heat Conduction in Finite Solids

[+] Author and Article Information
A. G. Ostrogorsky

Department of Mechanical Aerospace and Nuclear Engineering, and Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, NY 12211

J. Heat Transfer 131(1), 011303 (Oct 17, 2008) (11 pages) doi:10.1115/1.2977540 History: Received December 07, 2007; Revised June 10, 2008; Published October 17, 2008

## Abstract

Based on the one-term Fourier series solution, a simple equation is derived for low Biot number transient conduction in plates, cylinders, and spheres. In the $0 range, the solution gives approximately three times less error than the lumped capacity solution. For asymptotically low values of Bi, it approaches the lumped capacity solution. A set of equations valid for $0 is developed next. These equations are more involved but give approximately ten times lower error than the lumped capacity solution. Finally, a set of broad-range correlations is presented, covering the $0 range with less than 1% error.

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Copyright © 2009 by American Society of Mechanical Engineers
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## Figures

Figure 1

Transient temperature θ(x,t)∕θi in 2L plates calculated using Eqs. 20,21 together with the exact five-term Fourier series solution (full lines) and the lumped capacity solution (Eq. 4), dashed lines. (a) Bi=0.1. (b) Bi=0.3.

Figure 2

Transient temperature θ(r,t)∕θi in cylinders calculated using Eqs. 41,42, along with the exact five-term Fourier series solution (full lines) and the lumped capacity solution (Eq. 5), dashed lines. (a) Bi=0.1 (b) Bi=0.3.

Figure 3

Transient temperature θ(r,t)∕θi in spheres calculated using Eq. 55 along with the exact five term Fourier series solution (full lines) and the lumped capacity solution (Eq. 6), dashed lines. (a) Bi=0.1. (b) Bi=0.3.

Figure 4

Error: (a) B coefficients, error (%)=(Bexact−Bapprox)∕Bexact×100%; (b) Coefficients, error (%)=(Cexact−Capprox)∕Cexact×100%

Figure 5

Error=μexact−μapprox=Δμ1 as a function of the Biot number for (a) plate, (b) cylinder, and (c) sphere

Figure 6

Transient temperature θ(t)∕θi (at ξ=0), calculated using equations listed in Tables  45 (symbols). The full lines are the exact Fourier series solution. (a) Plate, (b) cylinder, and (c) sphere.

Figure 7

Error in midplane temperature, error (%)=(θexact−θapprox)∕θi×100%, for 2L plates as a function of Bi and Fo, calculated using equations listed in Table 4, with correlations for μ1 listed in (a) Table 5 and (b) Table 6.

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